Please explain the question number 3,4,6 and 8.
Mathematics Quiz
please explain question number 5 also.
- Six distinct positive integers are randomly chosen between 1 and 2006, inclusive. What is the probability that some pair of these integers has a difference that is a multiple of 5?
- From the set of numbers s={1, 2, 3, 4}, how many minimum numbers must be selected to guarantee that at least one pair of these numbers has a sum equal to 7?
- A dice is rolled 12 times independently. What is the expected value of the sum of the faces that rolled to the top?
- In how many ways 5 men and 3 women can sit together on a circle such that a particular men m1 and particular women w1 never sit adjacent to each other?
Hello @p8847220,
Answer 3: For two numbers to have a difference that is a multiple of 5, the numbers must be congruent \bmod{5} (their remainders after division by 5 must be the same).
0, 1, 2, 3, 4 are the possible values of numbers in \bmod{5}. Since there are only 5 possible values in \bmod{5} and we are picking 6 numbers, by the Pigeonhole Principle, two of the numbers must be congruent \bmod{5}.
Therefore the probability that some pair of the 6 integers has a difference that is a multiple of 5 is 1.
Answer 4: Since 3+ 4 add up to 7 so there exists a pair with sum 7
so minimum no. to be selected to get a pair with sum 7 is 4
Answer 8: refer to (3) in the below picture:
Answer 5: for n throws:
So, for 12 throws = 21*12/6 = 42
Hope, this would help.
thank you sir for your help